Merton model ($D_T=\min (V_T,F),E_T=(V_T-F{)}^{+}$)

In the basic structural model, the firm's asset value $V$ follows geometric Brownian motion with constant volatility and a single debt obligation of face value $F$ matures at $T$. Default at maturity occurs when $V_T<F$. Equity is a call on firm assets; debt is the risk-free face claim minus a put on those assets, since $\min (V_T,F)=F-(F-V_T{)}^{+}$. This links debt valuation to asset value, volatility and capital structure. The basic model does not permit default before maturity; richer structural models add a default barrier or more detailed obligations.